Over/Under
Exploration

Over/Under is an exploration of a simple tessellation concept: folding basic twists on both sides of the paper, at a density just below where pleats would overlap. This leads to both square tilings and equilateral triangle tilings. The exploration was done in March 2025.

After folding Shuzo Fujimoto's Galaxy tessellation (which features hexagon twists on one side and triangle twists on the other), and exploring the gallery of Madonna Yoder, I began to get a deeper appreciation of models that included twists on both sides of the paper. I decided to start exploring using the basic tessellation twists in a simple repeating pattern, to learn how they interact with each other and to get some practice.

The smallest twist, a closed twist (a relative size of 1) requires 3 pleats of spacing between twists. A medium twist, the open twist (a relative size of 2) requires 4 pleats, and the largest twist I explored, the double open twist (a relative size of 3), requires 5 pleats of spacing.

For square models, folding on a straight grid leads to an expanding diamond of twists. Folding on a rotated grid allows the pattern to cleanly reach the paper's edges.

Over/Under Square 1:1

The simplest Over/Under uses closed square twists on both sides of the paper. This produces a similar pattern on the front and back, with a slightly lower number of twists offset on the back. It has the densest spacing that creates a regular lattice pattern when backlit.

Crease pattern on a 32-square grid

The model folded on a 48-square grid (25cm double-sided origami paper)

Front side

Back side

Backlit front side

Backlit back side

The model folded on a 64-square rotated grid of ratio 3/8 (35cm origami paper)

Front side

Back side

Backlit front side

Backlit back side

Over/Under Square 1:2

The next step in Over/Under is to retain the closed square twist on the front but use open square twists on the back. This allows for a different view on the top and bottom sides. On the front side, the open square twists allow for an additional layer to appear "below".

Crease pattern on a 32-square grid

The model folded on a 48-square grid (25cm double-sided origami paper)

Front side

Back side

Backlit front side

Backlit back side

The model folded on a 64-square rotated grid of ratio 1/3 (35cm origami paper)

Front side

Back side

Backlit front side

Backlit back side

Over/Under Square 1:3

The 1:3 combination, with double open square twists on the back side, requires more space than the 1:2 combination. When viewed from the front, the sunken layer from the double open twist is quite large.

Crease pattern on a 32-square grid

The model folded on a 48-square grid (25cm double-sided origami paper)

Front side

Back side

Backlit front side

Backlit back side

The model folded on a 64-square rotated grid of ratio 3/10 (35cm origami paper)

Front side

Back side

Backlit front side

Backlit back side

Over/Under Square 2:2

Like the 1:1 model, the 2:2 model has a similar but offset pattern on both sides of the paper, but now have the sunken layer from the open twist appearing on both sides.

Due to the setup for open square twists, this is the only model that isn't perfectly centered.

Crease pattern on a 32-square grid

The model folded on a 48-square grid (25cm double-sided origami paper)

Front side

Back side

Backlit front side

Backlit back side

The model folded on a 64-square rotated grid of ratio 3/10 (35cm origami paper)

Front side

Back side

Backlit front side

Backlit back side

Over/Under Square 2:3

Like the 1:2 model, the 2:3 model provides a view with an upper tier of open twist being of a similar size to the sunken tier of the double open twist.

In order to align the model, the double open square twist is placed centrally behind the plane, connecting to open square twists above the plane.

Crease pattern on a 32-square grid

The model folded on a 48-square grid (25cm double-sided origami paper)

Front side

Back side

Backlit front side

Backlit back side

PENDING: This model can be folded on a rotated grid with ratio 3/11.

Over/Under Square 3:3

The 3:3 model is similar to the 1:1 and 2:2 models but requires a significant amount of space. This leads to a very airy lattice.

Crease pattern on a 32-square grid

The model folded on a 48-square grid (25cm double-sided origami paper)

Front side

Back side

Backlit front side

Backlit back side

PENDING: This model can be folded on a rotated grid with ratio 1/4.

Over/Under Triangle 1:1

The pattern can also be applied to triangle grids, where the twists surround a hexagonal center. This creates large star patterns in the negative space of the backlit model.